DLMF:13.4.E9 (Q4372): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1hdec

Latest revision as of 15:47, 2 January 2020

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DLMF:13.4.E9
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    Statements

    Defining formula
    𝐌 ⁡ ( a , b , z ) = Γ ⁡ ( 1 + a - b ) 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( a ) ⁢ ∫ 0 ( 1 + ) e z ⁢ t ⁢ t a - 1 ⁢ ( t - 1 ) b - a - 1 ⁢ d t , Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 Euler-Gamma 1 𝑎 𝑏 2 𝜋 imaginary-unit Euler-Gamma 𝑎 superscript subscript 0 limit-from 1 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑎 1 superscript 𝑡 1 𝑏 𝑎 1 𝑡 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E9
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    b - a ≠ 1 , 2 , 3 , … 𝑏 𝑎 1 2 3 … {\displaystyle{\displaystyle b-a\neq 1,2,3,\dots}}
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    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1agdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1hdec
    0 references
     
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